Motivated by [1]} and Remark REF , we solve Problem REF  by establishing a system of quasi-linear partial differential equations with Cauchy boundary data.
This system will be solved later by the method proposed in [2]}.
Let \(\lbrace \Psi _n\rbrace _{n \ge 1}\)  be an orthonormal basis of \(L^2(0, T).\) 
We approximate the solution \(u\)  to (REF ) by truncating its Fourier expansion with respect to the basis \(\lbrace \Psi _n\rbrace _{n \ge 1}\)  as follows
\(u({\bf x}, t) = \sum _{n = 1}^\infty u_n({\bf x}) \Psi _n(t)\simeq \sum _{n = 1}^N u_n({\bf x}) \Psi _n(t)\) 
